To allow a return to a pre-COVID-19 lifestyle, virtually every country has initiated a vaccination program to mitigate severe disease burden and control transmission; over 3.6 billion vaccine doses have been administered as of July 2021. However, it remains to be seen whether herd immunity will be within reach of these programs, especially as more transmissible SARS-CoV-2 variants continue to emerge. To address this question, we developed a data-driven model of SARS-CoV-2 transmission for Shanghai, China, a population with low prior immunity from natural infection. We found that extending the vaccination program to individuals aged 3-17 years plays a key role to reach herd immunity for the original SARS-CoV-2 lineages. With a vaccine efficacy 74% against infection, vaccine-induced herd immunity would require coverages of 93% or higher. Herd immunity for new variants, such as Alpha or Delta, can only be achieved with more efficacious vaccines and coverages above 80-90%. A continuation of the current pace of vaccination in China would reach 72% coverage by September 2021; although this program would fail to reach herd immunity it would reduce deaths by 95-100% in case of an outbreak. Efforts should be taken to increase population's confidence and willingness to be vaccinated and to guarantee highly efficacious vaccines against more transmissible variants of concern.